Bake and cool the crust as instructed on the recipe.
Beat eggs, egg yolks, and sugar with an electric mixer in a medium saucepan off the heat.
In a small bowl, dissolve the cornstarch and salt in a small amount of coconut milk. When no lumps remain, stir in the remainder of the coconut milk.
Combine the coconut milk mixture with the egg mixture in the saucepan and begin heating on the stovetop over a medium low heat. Stir continuously, scraping the bottom as you go. If the pudding begins to thicken at all on the surface of the pan, turn the heat down and continue stirring.
When the mixture thickens enough to coat the spoon and resembles loose pudding (about 10 minutes cook time), turn off the heat. If there are small lumps, you can use an immersion blender or hand mixer on a high speed to smooth out the custard. This is optional, as the tiny lumps won't impact the flavor.
Then add in the butter, one tablespoon at a time stirring by hand to combine each completely before adding the next. Finally stir in the vanilla by hand.
Transfer the custard to the baked and cooled pie shell. Place plastic wrap on top of the pie, flat against the custard to prevent a “skin” from forming on the custard as it chills. Store in the fridge until chilled through, which will take several hours or overnight.